v2006.03.09 - Convex Optimization

212 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS3.2.0.0.2 Definition. Quasiconvex function by sublevel sets.Matrix-valued function g(X) : R p×k →S M is a quasiconvex function ofmatrix X if and only if domg is a convex set and the sublevel setcorresponding to each and every V ∈ S M (confer (408))L V g = {X ∈ domg | g(X) ≼ V } ⊆ R p×k (437)is convex.△Necessity of convex sublevel sets is proven as follows: For any X, Y ∈ L V gwe must show for each and every µ ∈[0, 1] that µX + (1−µ)Y ∈ L V g . Bydefinition we have, for each and every real vector w of unit norm,w T g(µX + (1−µ)Y )w ≤ max{w T g(X)w , w T g(Y )w} ≤ w T V w (445)Likewise, convexity of all superlevel sets 3.11 is necessary and sufficient forquasiconcavity.3.2.1 Differentiable and quasiconvex3.2.1.0.1 Definition. Differentiable quasiconvex matrix-valued function.Assume function g(X) : R p×k →S M is twice differentiable, and domg is anopen convex set.Then g(X) is quasiconvex in X if wherever in its domain the directionalderivative (D.1.4) becomes 0, the second directional derivative (D.1.5) ispositive definite there [38,3.4.3] in the same direction Y ; id est, g isquasiconvex if for each and every point X ∈ dom g , all nonzero directionsY ∈ R p×k , and for t ∈ Rddt∣ g(X+ t Y ) = 0 ⇒t=03.11 The superlevel set is similarly defined:d2dt 2 ∣∣∣∣t=0g(X+ t Y ) ≻ 0 (446)L Sg = {X ∈ dom g | g(X) ≽ S }